Solid particles floating at a liquid interface exhibit a long-ranged attraction mediated by surface tension. In the absence of bulk elasticity, this is the dominant lateral interaction of mechanical origin. Here, we show that an analogous long-range interaction occurs between adjacent droplets on solid substrates, which crucially relies on a combination of capillarity and bulk elasticity. We experimentally observe the interaction between droplets on soft gels and provide a theoretical framework that quantitatively predicts the interaction force between the droplets. Remarkably, we find that, although on thick substrates the interaction is purely attractive and leads to dropdrop coalescence, for relatively thin substrates a short-range repulsion occurs, which prevents the two drops from coming into direct contact. This versatile interaction is the liquid-on-solid analog of the "Cheerios effect." The effect will strongly influence the condensation and coarsening of drops on soft polymer films, and has potential implications for colloidal assembly and mechanobiology.elastocapillarity | wetting | soft matter | mechanosensing | droplets T he long-ranged interaction between particles trapped at a fluid interface is exploited for the fabrication of microstructured materials via self-assembly and self-patterning (1-5) and occurs widely in the natural environment when living organisms or fine particles float on the surface of water (6, 7). In a certain class of capillary interactions, the particles deform the interface because of their shape or chemical heterogeneity (8)(9)(10). In this case, the change in interfacial area upon particle-particle approach causes an attractive capillary interaction between the particles. In the so-called Cheerios effect, the interaction between floating objects is mainly due to the change in gravitational potential energy associated to the weight of the particles, which deform the interface while being supported by surface tension (11), and the same principle applies when the interface is elastic (12-14). The name "Cheerios effect" is reminiscent of breakfast cereals floating on milk and sticking to each other or to the walls of the breakfast bowl.Here, we consider a situation opposite to that of the Cheerios effect, liquid drops deposited on a solid. The solid is sufficiently soft to be deformed by the surface tension of the drops, resulting in a lateral interaction. Recent studies have provided a detailed view of statics of single-drop wetting on deformable surfaces (15-19). The length scale over which the substrate is deformed is set by the ratio of the droplet surface tension γ and the substrate shear modulus G. The deformation can be seen as an elastocapillary meniscus, or "wetting ridge," around the drop (Fig. 1 A and B). Interestingly, the contact angles at the edge of the drop are governed by Neumann's law, just as for oil drops floating on water. In contrast to the statics of soft wetting, its dynamics has only been explored recently. New effects such as stick-slip motion induced...

We consider the dynamic snapping instability of elastic beams and shells. Using the Kirchhoff rod and Föppl-von Kármán plate equations, we study the stability, deformation modes, and snap-through dynamics of an elastic arch with clamped boundaries and subject to a concentrated load. For parameters typical of everyday and technological applications of snapping, we show that the stretchability of the arch plays a critical role in determining not only the postbuckling mode of deformation but also the timescale of snapping and the frequency of the arch's vibrations about its final equilibrium state. We show that the growth rate of the snap-through instability and its subsequent ringing frequency can both be interpreted physically as the result of a sound wave in the material propagating over a distance comparable to the length of the arch. Finally, we extend our analysis of the ringing frequency of indented arches to understand the 'pop' heard when everted shell structures snap-through to their stable state. Remarkably, we find that not only are the scaling laws for the ringing frequencies in these two scenarios identical but also the respective prefactors are numerically close; this allows us to develop a master curve for the frequency of ringing in snapping beams and shells. ULC 0.5 N load cell. p-2

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